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38
New Multiscale Transforms, Minimum Total Variation Synthesis: Applications to EdgePreserving Image Reconstruction
, 2001
"... This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and acc ..."
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Cited by 79 (11 self)
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This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘denoising.’
The Finite Ridgelet Transform for Image Representation
 IEEE Transactions on Image Processing
, 2003
"... The ridgelet transform [6] was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we propose an orthonormal version of the ridgelet transform for discrete and finite size images. Our construction uses the finite ..."
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Cited by 75 (2 self)
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The ridgelet transform [6] was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we propose an orthonormal version of the ridgelet transform for discrete and finite size images. Our construction uses the finite Radon transform (FRAT) [11], [20] as a building block. To overcome the periodization effect of a finite transform, we introduce a novel ordering of the FRAT coefficients. We also analyze the FRAT as a frame operator and derive the exact frame bounds. The resulting finite ridgelet transform (FRIT) is invertible, nonredundant and computed via fast algorithms. Furthermore, this construction leads to a family of directional and orthonormal bases for images. Numerical results show that the FRIT is more effective than the wavelet transform in approximating and denoising images with straight edges.
Sparse Directional Image Representations using the Discrete Shearlet Transform
 Appl. Comput. Harmon. Anal
"... It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a n ..."
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Cited by 48 (28 self)
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It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications both in terms of performance and computational efficiency.
Random sampling of multivariate trigonometric polynomials
 SIAM J. Math. Anal
, 2004
"... We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for th ..."
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Cited by 30 (3 self)
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We investigate when a trigonometric polynomial p of degree M in d variables is uniquely determined by its sampled values p(xj) on a random set of points xj in the unit cube (the “sampling problem for trigonometric polynomials”) and estimate the probability distribution of the condition number for the associated Vandermondetype and Toeplitzlike matrices. The results provide a solid theoretical foundation for some efficient numerical algorithms that are already in use.
Nearoptimal detection of geometric objects by fast multiscale methods
 IEEE Trans. Inform. Theory
, 2005
"... Abstract—We construct detectors for “geometric ” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detec ..."
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Cited by 23 (7 self)
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Abstract—We construct detectors for “geometric ” objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in twodimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detection threshold—i.e., the signal strength below which no method of detection can be successful for large dataset size. ii) The optimal computational complexity of a nearoptimal detector, i.e., the complexity required to detect signals slightly exceeding the detection threshold. We describe a general approach to such problems which covers several classes of geometrically defined signals; for example, with onedimensional data, signals having elevated mean on an interval, and, indimensional data, signals with elevated mean on a rectangle, a ball, or an ellipsoid. In all these problems, we show that a naive or straightforward approach leads to detector thresholds and algorithms which are asymptotically far away from optimal. At the same time, a multiscale geometric analysis of these classes of objects allows us to derive asymptotically optimal detection thresholds and fast algorithms for nearoptimal detectors. Index Terms—Beamlets, detecting hot spots, detecting line segments, Hough transform, image processing, maxima of Gaussian processes, multiscale geometric analysis, Radon transform. I.
A framework for discrete integral transformations II – the 2D 31 Radon transform
"... This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopola ..."
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Cited by 21 (10 self)
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This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopolar Fourier transform that samples the Fourier transform on the pseudopolar grid, also known as the concentric squares grid. The pseudopolar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudopolar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only
Wavelets, Ridgelets, and Curvelets for Poisson Noise Removal
"... Abstract—In order to denoise Poisson count data, we introduce a variance stabilizing transform (VST) applied on a filtered discrete Poisson process, yielding a near Gaussian process with asymptotic constant variance. This new transform, which can be deemed as an extension of the Anscombe transform t ..."
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Cited by 18 (1 self)
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Abstract—In order to denoise Poisson count data, we introduce a variance stabilizing transform (VST) applied on a filtered discrete Poisson process, yielding a near Gaussian process with asymptotic constant variance. This new transform, which can be deemed as an extension of the Anscombe transform to filtered data, is simple, fast, and efficient in (very) lowcount situations. We combine this VST with the filter banks of wavelets, ridgelets and curvelets, leading to multiscale VSTs (MSVSTs) and nonlinear decomposition schemes. By doing so, the noisecontaminated coefficients of these MSVSTmodified transforms are asymptotically normally distributed with known variances. A classical hypothesistesting framework is adopted to detect the significant coefficients, and a sparsitydriven iterative scheme reconstructs properly the final estimate. A range of examples show the power of this MSVST approach for recovering important structures of various morphologies in (very) lowcount images. These results also demonstrate that the MSVST approach is competitive relative to many existing denoising methods. Index Terms—Curvelets, filtered Poisson process, multiscale variance stabilizing transform, Poisson intensity estimation, ridgelets, wavelets. I.
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 17 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Fast Xray and beamlet transforms for threedimensional data
 in Modern Signal Processing
, 2002
"... Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop t ..."
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Cited by 12 (8 self)
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Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop tools for the analysis of 3D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the Xray transform, offering the collection of line integrals along a wide range of lines running through the image — at all different orientations and positions; and (b) Multiscale: the (3D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data. 1.
On the computation of the polar FFT
 Appl. Comput. Harmon. Anal
, 2007
"... We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT. ..."
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Cited by 9 (7 self)
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We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.